An Approximate Analytical Solution for Counter-Current Spontaneous Imbibition

An approximate analytical solution is provided for one-dimensional, counter- current, spontaneous imbibition of a wetting phase (water) into a semi-infinite porous medium. The solution is based on the assumption that a similarity solution exists for the displacement process. This assumption, in turn, rests on the assumption that the set of relative permeability and capillary pressures curves are unique functions of saturation and do not depend on the nature of the displacement. It further rests on the assumption that the saturation at the imbibition face does not vary with time. It is demonstrated that the solution is in agreement with results obtained from experiments and also numerical analyses of these experiments. The experiments utilize cylindrical samples with the radial surface and one end-face sealed, and with counter-current imbibition occurring at the open end-face. The stage of the experiment that is modeled by the present solution is the period before the imbibition front contacts the sealed end-face. An important finding of the present analysis is that the pressure upstream of the advancing invasion front is a constant. A second, improved solution is also presented; this solution is an iterative, series solution of an integral-differential equation. It converges to a stable solution in very few terms.